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Models with limited dependent variables

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Models with limited dependent variables

Doctoral Program 2006-2007

Katia Campo

3. Nested Logit Model

(K.Train, Ch.4, Franses and Paap Ch.5)

- Choice between J>2 alternatives which can be grouped into subsets based on differences in substitution pattern
- Example:

Choice of transportation mode

Car

Transit

Carpool

Bus

Train

Car alone

- Assumption cumulative distribution of error terms
- Probability

- Probability can be decomposed into 2 logit models

(1)

(2)

- Link between Pni | Bk * Pn Bk (upper and lower model level) through inclusive value Ink
- To comply with RUM k must be in the [0,1] interval
- IIA within, not across nests

Estimation

- Joint estimation
- Sequential estimation
- Estimate lower model
- Compute inclusive value
- Estimate upper model with inclusive value as explanatory variable
Disadvantages sequential estimation

- Add fourth step, using the parameter estimates as starting values for joint estimation

- Example: Location choices by French firms in Eastern and Western Europe
- Dependent var.: probability of choosing location j Pj=P(πj> πk k≠j )
- Location choices are likely to have a nested structure (non-IIA)
- First, select region (East or Western Europe)
- Next, select country within region

Disdier and Mayer (2004)

- Example 1: Location choices by French firms in Eastern and Western Europe

Location choice

E.Eur

W.Eur

………

………

CN

C1

CJ+1

CJ

Disdier and Mayer (2004)

- Location choices: Data
- 1843 location decisions in Europe from 1980 to 1999 (official statistics)
- 19 host countries (13 W.Eur, 6 E.Eur)

Disdier and Mayer (2004)

- Location choices: Data
- NFFrench firms already located in the country
- GDPGDP
- GPP/CAPGDP per capita
- DISTDistance France – host country
- WAverage wage per capita (manufacturing)
- UNEMPLunemployment rate
- EXCHRExchange rate volatility
- FREEFree country
- PNFREEPartly free and not free country
- PR1Country with political rights rated 1
- PR2Country with political rights rated 2
- PR345Country with political rights rated 3,4,5
- PR67Country with political rights rated 6,7
- LIAnnual liberalization index
- CLICumulative liberalization index
- ASSOC=1 if an association agreement is signed

Disdier and Mayer (2004)

3. Nested Logit Model

Disdier and Mayer (2004)

3. Nested Logit Model

Disdier and Mayer (2004)

3. Nested Logit Model

- Example 2: Shopping centre choice

3. Nested Logit Model

- Example 2: Shopping centre choice

3. Nested Logit Model

- Example 2: Shopping centre choice

- (Purchase) incidence and choice models can be linked in the same way

Include incl.value

Purchase decision

No purchase

Purchase

...

(See above: Bucklin & Gupta)

Altern. J

Altern.1

- Other examples
- Private label versus nationals brands
- Same versus other brand
- Fixed rate (low risk) versus variable rate (high risk) investments
- Choice of transportation mode
- ....

4. Probit Model

(K.Train, Ch.5 ; Franses and Paap, Ch.4-5)

- Based on the general RUM-model
- Ass.: error terms are distributed normal with a mean vector of zero and covariance matrix
- density function:

Logit or Probit?

- Trade-off between tractability and flexibility
- Closed-form expression of the integral for Logit, not for Probit models
- Probit allows for random taste variation, can capture any substitution pattern, allows for correlated error terms and unequal error variances

Dependent on the specifics of the choice situation

Estimation

- Approximation of the multidimensional integral
- Non-simulation procedures (see Kamakura 1989)
Can usually only be applied to restricted cases and/or provide inaccurate estimations

- Simulation procedures (see Geweke et al.1994, Train)

- Non-simulation procedures (see Kamakura 1989)

Simulation-based estimation(binary probit, CFS)

- Step 1
- For each observation n=1, ..., N draw r ~ N(0,1), (r = 1, ......., R: repetitions)
- Initialize y_count= 0, =mt (starting values)
- Compute y*rn = xn mt + L r ; L= choleski factor (LL’= )
- Evaluate: y*rn >0 y_count= y_count+1
- Repeat R times

Weeks (1997)

Simulation-based estimation(binary probit, CFS)

- Step 2: calculate probabilities
Pn| mt= y_count/R

- Step 3: Form the simulated LL function
SLL= n yn ln(Pn|mt)+(1-yn) ln(1-Pn|mt)

- Step 4: Check convergence criteria (SLL(mt)- SLL(mt-1))
- Step 5: Update mt: mt+1 = mt + v
- Step 6: Iterate (until convergence)

Weeks (1997)

Simulation-based estimation: MNProbit

- Based on same principles
- More efficient simulation procedures (see Train)
- Identification: normalization of level and scale
- Re-express model in utility differences
- Normalization of varcov matrix (see Train)

- Random taste variation
- Model with random coefficients
- E.g.:n~N(b,W)
- Unj = b’xnj+ *’n xnj + nj
- = b’xnj+ nj
- nj : correlated error terms (dep.on xnj, see Train)

- Substitution patterns
- Full covariance matrix (no parameter restrictions) unrestricted substitution patterns
- Structured covariance matrix (restrictions on some covariance parameters)
- the structure imposed on determines the substitution pattern and may allow to reduce the number of parameters to be estimated

- Example (Kamakura and Srivastava 1984): random utility components ni, nj are more (less) highly correlated when i and j are more (less) similar on important attributes

(dij = weighted eucledian distance between i & j)

- Examples
- Choice models at brand-size level: correlation between ≠ sizes of same brand (Chintagunta 1992)

MNL model

gives biased

estimates

of price

elasticity

- Examples
- Firm innovation (Harris et al. 2003)
- Binary probit model for innovative status (innovation occurred or not)
- Based on panel data correlation of innovative status over time: unobserved heterogeneity related to management ability and/or strategy

- Firm innovation (Harris et al. 2003)

Model (2)-(4) account for unobserved heterogeneity (ρ) -> superior results

- Examples
- Dynamics of individual health (Contoyannis, Jones and Nigel 2004)
- Binary probit model for health status (healthy or not)
- Survey data for several years correlation over time (state dependence) + individual-specific (time-invariant) random coefficient

- Dynamics of individual health (Contoyannis, Jones and Nigel 2004)

- Examples
- Choice of transportation mode (Linardakis and Dellaportas 2003) Non-IIA substitution patterns

5. Ordered Logit Model

- Choice between J>2 ordered ‘alternatives’
- Ordinal dependent variable y = 1, 2, ... J, with
rank(1) < rank(2) < ... < rank(J)

- Example:
- Purchase of 1, 2, ... J units
- Evaluation on a J-point scale ranging from, e.g., ‘dislike very much’ to ‘like very much’

- Suppose yi* is a continous latent variable which
- is a linear function of the explanatory variables
yn* = Xn + n

- and can be ‘mapped’ on an ordered multinomial variable as follows:
yn= 1 if 0 < yn* 1

yn= j if j-1 < yn* j

yn= J if J-1 < yn* J

0 < 1 < …. < j < … < J

- is a linear function of the explanatory variables

Ordered logit (see above)

- 0 , J and 0: set equal to zero

Interpretation of parameters (marginal effects)

Estimation: ML

- Disadvantages (Borooah 2002)
- Assumption of equal slope k
- Biased estimates if assumption of strictly ordered
outcomes does not hold

- treat outcomes as nonordered unless there are
good reasons for imposing a ranking

Example: Effectiveness of better public transit as a way to reduce automobile congestion and air polution in urban areas

- Research objective: develop and estimate models to measure how public transit affects automobile ownership and miles driven.
- Data: Nationwide Personal Transportation Survey (42.033 hh): socio-demo’s, automobile ownership and use, public transportation avail.

Kim and Kim (2004)

- Dependent variable ownership model = number of cars (k = 0, 1, 2, 3) ordinal variable
- C*i = latent variable: automobile ownership propensity of hh i
- Relation to observed automobile ownership:
Ci=k if k-1 < ’xi + < k

- P(Ci=k)=F(k- ’xi) - F(k-1 - ’xi)

Kim and Kim (2004)

5. Ordered Logit Model

5. Ordered Logit Model

5. Ordered Logit Model

5. Ordered Logit Model

- Examples
- Occupational outcome as a function of socio-demographic characteristics (Borooah)
- Unskilled/semiskilled
- Skilled manual/non-manual
- Professional/managerial/technical

- School performance (Sawkins 2002)
- Grade 1 to 5
- Function of school, teacher and student characteristics

- Level of insurance coverage

- Observed heterogeneity
- Unobserved heterogeneity
- Over decision makers
- Random coefficients Models
- E.g. Mixed Logit Model (see Train)

- Over segments
- Latent class estimation

- Over decision makers

- Ass.: Consumers can be placed into a small number of – homogeneous - segments which differ in choice behavior ( response parameters)
- Relative size of the segment s (s=1, 2, ..., M) is given by
fs = exp(s) / s’exp(s’)

Kamakura and Russell (1989)

- Probability of choosing brand j, conditional on consumer i being a member of segment s
Ps(yn = j|Xn) = exp(Xjns)

lexp(Xlns)

- Unconditional probability that consumer i will choose brand j
P(yn = j|Xn) = sfs Ps(yn = j|Xn)

= s exp(s) exp(Xjns)

s’exp(s’) lexp(Xlns)

- Estimation: Maximum Likelihood
- Likelihood of a hh’s choice history Hn
L(Hn) = s [ exp(s)L(Hn|s) / s’ exp(s’) ]

with

L(Hn|s) = t Ps(ynt = c(t) | Xnt)

c(t) = index of the chosen option at time t.

- Maximize likelihood over all hh’s: n L(Hn)

Segment analysis

- Based on parameter estimates (e.g. difference in price sensitivity)
- Based on segment profiles
- Post-hoc: based on assignment of hh to segments; Probability that hh n belongs to segment s =
P(ns | Hn) = L(Hn|s)fs / s’ [L(Hn|s’)fs’]

Analyze characteristics of different segments

- A priori: make fs a function of variables that may explain segment membership (e.g. income for segments which differ in price sensitivity)

- Post-hoc: based on assignment of hh to segments; Probability that hh n belongs to segment s =

- Example: heterogeneity in price sensitivity (Bucklin and Gupta 1992)

- Example: heterogeneity in price sensitivity

Choice segments: segment 1 = more sensitive to price and promo

Incidence segments: segment 2 and 4 = more sensitive to changes

in category attractiveness (change in price/promo)

Confirms that combinations of choice/inc.price sensitivity occur

- Segment analysis: price elasticity

- Segment analysis: socio-demographic profile